# The Church of the larger Hilbert space

## Introduction

John Smolin coined the phrase "Going to the Church of the Larger Hilbert Space" for the dilation constructions of channels and states, which not only provide a neat characterization of the set of permissible quantum operations but are also a most useful tool in quantum information science.

According to Stinespring's dilation theorem, every completely positive and trace-preserving map, or channel, can be built from the basic operations of (1) tensoring with a second system in a specified state, (2) unitary transformation, and (3) reduction to a subsystem. Thus, any quantum operation can be thought of as arising from a unitary evolution on a larger (dilated) system. The auxiliary system to which one has to couple the given one is usually called the ancilla of the channel. Stinespring's representation comes with a bound on the dimension of the ancilla system, and is unique up to unitary equivalence.

## Stinespring's dilation theorem

We present Stinespring's theorem in a version adapted to completely positive and trace-preserving maps between finite-dimensional quantum systems. For simplicity, we assume that the input and output systems coincide. The theorem applies more generally to completely positive (not necessarily trace-preserving) maps between C * algebras.

Stinespring's dilation: Let $T : S(\mathcal{H}) \rightarrow S(\mathcal{H})$ be a completely positive and trace-preserving map between states on a finite-dimensional Hilbert space H. Then there exists a Hilbert space $\mathcal{K}$ and a unitary operation U on $\mathcal{H} \otimes \mathcal{K}$ such that
$T(\varrho) = tr_{\mathcal{K}} U^{}( \varrho \otimes |0\rangle \langle 0|)U^{\dagger}$
for all $\varrho \in S(\mathcal{H})$, where $tr_{\mathcal{K}}$ denotes the partial trace on the $\mathcal{K}-$system.
The ancilla space $\mathcal{K}$ can be chosen such that $\dim \mathcal{K} \leq \dim^{2} \mathcal{H}$. This representation is unique up to unitary equivalence.

## Kraus decomposition

It is sometimes useful not to go to a larger Hilbert space, but to work with operators between the input and output Hilbert spaces of the channel itself. Such a representation can be immediately obtained from Stinespring's theorem: We introduce a basis $|k\rangle$ of the ancilla space $\mathcal{K}$ and define the Kraus operators tk in terms of Stinespring's unitary U as

$\langle a|t_k|b \rangle := \langle a \otimes k |U|b \otimes 0 \rangle$

The Stinespring representation then becomes the operator-sum decomposition or Kraus decomposition of the quantum channel T:

Kraus decomposition: Every completely positive and trace-preserving map $T : S(\mathcal{H}) \rightarrow S(\mathcal{H})$ can be given the form
$T(\varrho) = \sum_{k=1}^{K} t_{k}^{} \, \varrho \, t_{k}^{\dagger}$
for all $\varrho \in S(\mathcal{H})$. The $K \leq \dim^{2} \mathcal{H}$ Kraus operators $t_k : \mathcal{H} \rightarrow \mathcal{H}$ satisfy the completeness relation $\sum_{k} t_{k}^{\dagger} t_{k}^{} = \mathbf{1}$.

## Purification of quantum states

Quantum states are channels $\varrho: \mathbb{C} \rightarrow S(\mathcal{H})$ with one-dimensional input space $\mathbb{C}$ (cf. Channel (CP map)). We may thus apply Stinespring's dilation theorem to conclude that $\varrho$ can be given the representation

$\varrho = tr_{\mathcal{K}} |\psi\rangle\langle \psi |$,

where $|\psi \rangle = U |0 \rangle$ is a pure state on the combined system $\mathcal{H} \otimes \mathcal{K}$. In other words, every mixed state $\varrho$ can be thought of as arising from a pure state $|\psi \rangle$ on a larger Hilbert space. This special version of Stinespring's theorem is usually called the GNS construction of quantum states, after Gelfand and Naimark, and Segal.

For a given mixed state with spectral decomposition $\varrho = \sum_k p_k \, |k\rangle \langle k| \, \in S(\mathcal{H})$, such a purification is given by the state

$|\psi \rangle = \sum_k \, \sqrt{p_k} \, |k \rangle \otimes |k\rangle \, \in \mathcal{H} \otimes \mathcal{H}$.